Question: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$2.50$ each for students, and the group paid $$35.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$24.00$ each for teachers and $$9.50$ each for students, and the group paid $$138.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+2.5y = 35.5}$ ${24x+9.5y = 138.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-24x-10y = -142}$ ${24x+9.5y = 138.5}$ Add the top and bottom equations together. $ -0.5y = -3.5 $ $ y = \dfrac{-3.5}{-0.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {6x+2.5y = 35.5}$ to find $x$ ${6x + 2.5}{(7)}{= 35.5}$ $6x+17.5 = 35.5$ $6x = 18$ $x = \dfrac{18}{6}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {24x+9.5y = 138.5}$ and get the same answer for $x$ ${24x + 9.5}{(7)}{= 138.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.